In this paper, we show the relation between the Schur algebras S r Λ,Σ (β) and S r′ Λ,Σ (β); where 1 ≤ r′ < r < ∞. Then we set up the involution operator in these Schur algebras and show that with this involution operator there is only one C*-algebra among these classes of Banach algebras. Furthermore, we show the equivalence of a condition on the Schur multiplier norm and the existence of C*-algebra.
